Hartree atomic orbital

The Pauli exclusion principle states that no two electrons in an atom can have the same set of four quantum numbers (n, , m, wrj. Another way of stating this principle is that each Hartree atomic orbital (characterized by a set of three quantum numbers, n, , and m) holds at most two electrons, one with spin up and the other with spin down. 

Linear Combination of Atomic Orbitals (LCAO) in Hartree-Fock Theory... 

Ihe one-electron orbitals are commonly called basis functions and often correspond to he atomic orbitals. We will label the basis functions with the Greek letters n, v, A and a. n the case of Equation (2.144) there are K basis functions and we should therefore xpect to derive a total of K molecular orbitals (although not all of these will necessarily 3e occupied by electrons). The smallest number of basis functions for a molecular system vill be that which can just accommodate all the electrons in the molecule. More sophisti- ated calculations use more basis functions than a minimal set. At the Hartree-Fock limit he energy of the system can be reduced no further by the addition of any more basis unctions however, it may be possible to lower the energy below the Hartree-Fock limit ay using a functional form of the wavefunction that is more extensive than the single Slater determinant. 

We now introduce the atomic orbital expansion for the orbitals i/), and substitute for the corresponding spin orbital Xi into the Hartree-Fock equation,/,(l)x,(l) = X (1) ... 

Having the Slater atomic orbitals, the linear combination approximation to molecular orbitals, and the SCF method as applied to the Fock matrix, we are in a position to calculate properties of atoms and molecules ab initio, at the Hartree-Fock level of accuracy. Before doing that, however, we shall continue in the spirit of semiempirical calculations by postponing the ab initio method to Chapter 10 and invoking a rather sophisticated set of approximations and empirical substitutions... 

The second approximation in HF calculations is due to the fact that the wave function must be described by some mathematical function, which is known exactly for only a few one-electron systems. The functions used most often are linear combinations of Gaussian-type orbitals exp(—nr ), abbreviated GTO. The wave function is formed from linear combinations of atomic orbitals or, stated more correctly, from linear combinations of basis functions. Because of this approximation, most HF calculations give a computed energy greater than the Hartree-Fock limit. The exact set of basis functions used is often specified by an abbreviation, such as STO—3G or 6—311++g. Basis sets are discussed further in Chapters 10 and 28. 

Configuration Interaction (or electron correlation) adds to the single determinant of the Hartree-Fock wave function a linear combination of determinants that play the role of atomic orbitals. This is similar to constructing a molecular orbital as a linear combination of atomic orbitals. Like the LCAO approximation. Cl calculations determine the weighting of each determinant to produce the lowest energy ground state (see SCFTechnique on page 43). 

But alas most of what has been described so far concerning density theory applies in theory rather than in practice. The fact that the Thomas-Fermi method is capable of yielding a universal solution for all atoms in the periodic table is a potentially attractive feature but is generally not realized in practice. The attempts to implement the ideas originally due to Thomas and Fermi have not quite materialized. This has meant a return to the need to solve a number of equations separately for each individual atom as one does in the Hartree-Fock method and other ab initio methods using atomic orbitals. 

Table 5.1 Effect of relativity on Hartree-Eock orbital energies (in eV) for the neutral Hg and Fe atoms. Scalar relativistic effects were treated with the DKH2 approximation...

Rauhut, G., Puyear, S., Wolinski, K., Pulay, P., 1996, Comparison of NMR Shielding Calculated from Hartree-Fock and Density Functional Wave Functions Using Gauge-Including Atomic Orbitals , J. Phys. Chem., 100,... 

The Hartree-Fock orbitals are expanded in an infinite series of known basis functions. For instance, in diatomic molecules, certain two-center functions of elliptic coordinates are employed. In practice, a limited number of appropriate atomic orbitals (AO) is adopted as the basis. Such an approach has been developed by Roothaan 10>. In this case the Hartree-Fock differential equations are replaced by a set of nonlinear simultaneous equations in which the limited number of AO coefficients in the linear combinations are unknown variables. The orbital energies and the AO coefficients are obtained by solving the Fock-Roothaan secular equations by an iterative method. This is the procedure of the Roothaan LCAO (linear-combination-of-atomic-orbitals) SCF (self-consistent-field) method. 

Experimentally determined maximum absolute ionization cross sections for the inert gases and a range of small molecules are compared with the predictions of DM, BEB, and EM calculations in Table 1. Atomic orbital coefficients for the DM calculations were determined at the Hartree-Fock level and the EM cross sections are volume averaged for calculations carried out at the HF/6-31G level. Hie same data are plotted in Figure 5 with the calculated values on the ordinate and the experimental result on the abscissa. The heavy line represents a direct correspondence between experiment and theory. Although the ab initio EM method performs well for the calculation of qm and Em,T,17 the DM and BEB methods allow for the calculation of the cross section as a function of the electron energy, i.e. the ionization... 

When the Hartree-Fock method is applied to molecules, molecular orbitals are used instead of atomic orbitals. To construct the molecular orbitals, one widely used approximation is LCAO (linear combinations of atomic orbitals). According to molecular orbital theory, the total wave function of the system is written as a combination of molecular orbitals, spin functions describing electrons in terms of spin j(a) or — j p). 

P has been computed using Hartree-Fock atomic orbital wavefunctions and can be found in several published tabulations14 17 and in Appendix 1. Because of the (r 3) dependence of P, dipolar coupling of a nuclear spin with electron spin density on another atom is usually negligible. 

Most of the commonly used electronic-structure methods are based upon Hartree-Fock theory, with electron correlation sometimes included in various ways (Slater, 1974). Typically one begins with a many-electron wave function comprised of one or several Slater determinants and takes the one-electron wave functions to be molecular orbitals (MO s) in the form of linear combinations of atomic orbitals (LCAO s) (An alternative approach, the generalized valence-bond method (see, for example, Schultz and Messmer, 1986), has been used in a few cases but has not been widely applied to defect problems.)... 

The calculation of the indices requires the overlap matrix S of atomic orbitals and the first-order density (or population) matrix P (in open-shell systems in addition the spin density matrix Ps). The summations refer to all atomic orbitals /jl centered on atom A, etc. These matrices are all computed during the Hartree-Fock iteration that determines the molecular orbitals. As a result, the three indices can be obtained... 

Although P3 procedures perform well for a variety of atomic and molecular species, caution is necessary when applying this method to open-shell reference states. Systems with broken symmetry in unrestricted Hartree-Fock orbitals should be avoided. Systems with high multireference character are unlikely to be described well by the P3 or any other diagonal approximation. In such cases, a renormalized elec-... 

The first calculations on a two-electron bond was undertaken by Heitler and London for the H2 molecule and led to what is known as the valence bond approach. While the valence bond approach gained general acceptance in the chemical community, Robert S. Mulliken and others developed the molecular orbital approach for solving the electronic structure problem for molecules. The molecular orbital approach for molecules is the analogue of the atomic orbital approach for atoms. Each electron is subject to the electric field created by the nuclei plus that of the other electrons. Thus, one was led to a Hartree-Fock approach for molecules just as one had been for atoms. The molecular orbitals were written as linear combinations of atomic orbitals (i.e. hydrogen atom type atomic orbitals). The integrals that needed to be calculated presented great difficulty and the computations needed were... 

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